Respuesta :
Answer:
1.82 inches
Step-by-step explanation:
Let the size of square piece cut from the corner is d.
So, the length of the box is 15 - 2y
width of the box is 9 - 2y
height of the box is d.
Volume of the box
Volume = length x width x height
V = (15 - 2y)(9 - 2y)y
V = 4y³ - 48y² + 135y
Differentiate with respect to y.
[tex]\frac{dV}{dy}=12y^{2}-96y+135[/tex]
It is equal to zero for maxima and minima
[tex]12y^{2}-96y+135=0[/tex]
[tex]y=\frac{96\pm \sqrt{96^{2}-4\times 12\times 135}}{24}[/tex]
So, y = 1.82 in or 6.2 in
Now
[tex]\frac{d^{2}{V}}{dy^{2}}=24y-96[/tex]
For y = 1.82 in, it is negative
and for y = 6.2 in, it is positive
So, the volume of the box is maximum if the height of the box is 1.82 inch.
Thus, the size of the square cut from each of the corner is 1.82 inches.
